Let's solve the problem step by step.

Step 1: Define the variables

Let:

• $d$ be the distance between cities A and B.
• $v$ be the initial speed in km/h.
• $t$ be the time (in hours) it would take Sam to reach city B at the initial speed.

Step 2: Set up the equations

1. Given Information:

   • Sam drives 80 km in 2 hours. So, the initial speed $v$ is: $v = \frac{80 \text{ km}}{2 \text{ hours}} = 40 \text{ km/h}$
   • If Sam continued at the same speed, she would be 15 minutes (or $\frac{1}{4}$ hours) late. Therefore, the actual time at this speed would be $t + \frac{1}{4}$.

2. Equation for Distance with Initial Speed:

   • Distance $d$ using the initial speed: $d = v \times t = 40 \times t$

3. Time Taken After Increasing Speed:

   • When Sam increases her speed by 10 km/h, her new speed becomes $40 + 10 = 50$ km/h.
   • With this new speed, she arrives 36 minutes (or $\frac{3}{5}$ hours) earlier than planned.
   • The time with the increased speed is $t - \frac{3}{5}$.
   • Distance $d$ with the new speed: $d = 50 \times \left( t - \frac{3}{5} \right)$

Step 3: Equating the two expressions for distance

Since both expressions represent the same distance $d$, we can set them equal to each other: $40t = 50 \left( t - \frac{3}{5} \right)$

Step 4: Solve for $t$

Expanding and solving the equation: $40t = 50t - 30$ $10t = 30$ $t = 3 \text{ hours}$

Step 5: Calculate the distance $d$

Now, use the value of $t$ to find $d$: $d = 40 \times t = 40 \times 3 = 120 \text{ km}$

Final Answer

The distance between cities A and B is $\boxed{120 \text{ km}}$.

Would you like more details, or do you have any questions?

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Here are 5 related questions to expand on this topic:

1. How would the problem change if Sam increased her speed by 20 km/h instead of 10 km/h?
2. What if Sam had decreased her speed instead of increasing it? How would that affect the arrival time?
3. How does the concept of average speed play into solving this type of problem?
4. If Sam had to make a stop halfway between the two cities, how would that affect the overall time and distance calculations?
5. What other factors, such as traffic conditions or road types, could impact the calculation of speed and distance?

Tip: Always double-check your units (e.g., time in hours, speed in km/h) to ensure consistency in calculations.